Vincent Wesley Couey

Vincent Wesley Couey

Many engineering failures (thermal hotspot concentration, Hertz contact fatigue localization, boundary-layer loss, mixing dead zones) are geometric failure modes: changing the material delays the failure; changing the geometry eliminates it. Despite this, no formal metric exists for evaluating how uniformly a convex body distributes surface contact during rolling, with direct engineering implications. We introduce the Contact Distribution Score (CDS), a scalar metric defined as the area-weighted variance of contact time over a rolling surface, and its stress-domain counterpart the Stress Distribution Score (SDS), the area-weighted variance of accumulated Hertz contact pressure. CDS -> 0 indicates uniform contact; SDS -> 0 indicates uniform stress. We implement a three-layer oracle architecture (approximate oracle for search, rigid-body oracle for validation, Hertz contact pressure oracle for SDS). A parametric search over 45 members of the developable roller family identifies the oloid (Schatz, 1929) at CDS = 8.2 x 10^-7, with the conventional cylinder baseline at 4.75 x 10^-5: a 58x discrimination. Independent curvature-driven analysis under uniform contact yields a geometry-only SDS of 4.8 x 10^-8, indicating the oloid's surface curvature contributes minimal additional stress non-uniformity beyond the contact distribution. We extend the analysis to fatigue (FDS), thermal (TDS), and wear (WDS) scores, finding the oloid's 58x advantage transfers consistently across linear and multiplicative metrics in a 46-68x range. The nonlinear fatigue metric diverges due to Basquin S-N amplification but still shows oloid superiority over all tested alternatives. This work establishes the formal vocabulary and computational infrastructure for substrate geometry: the study of geometric forms as engineering substrates classified by their operational invariants.

Vincent Wesley Couey

Many engineering failures in orientation-dependent systems are geometric failure modes: changing the geometry can eliminate what changing the material merely delays. The mono-monostatic property (exactly one stable equilibrium under gravity) is mathematically proven to exist in convex homogeneous bodies, but no verified geometry has been openly published. We introduce an Equilibrium Count Score (ECS) oracle measuring stable equilibria via drainage basin analysis on the center-of-mass height landscape. Applying this oracle to Sloan's (2023) analytical Gomboc parameterization, we find that no tested parameter value produces a mono-monostatic body. The surface function has two critical points as proven, but the COM height landscape exhibits 4-11 local minima. Surface critical points are necessary but not sufficient for mono-monostatic behavior. We close this gap by extending the Sloan phase function with Fourier terms and optimizing via differential evolution, constructing three verified mono-monostatic bodies with ECS=1 confirmed across merge thresholds from 0.5% to 10%. The primary instance (beta=0.023, a1=0.234) is the first openly published, computationally verified mono-monostatic geometry. The central result: conventional geometries cannot achieve ECS=1 through ballast alone. Cylinders retain multiple equilibria even at 30% bottom-weighted mass. Applied to IMU calibration housing (349x precision improvement, zero prior art), aerial reforestation seed pods (eliminating 20-67% germination loss from orientation), and marine buoy self-righting. Cross-layer scoring confirms the Gomboc is 11.8x worse than the cylinder on contact distribution while optimal on equilibrium stability, demonstrating framework discrimination across three invariant classes.

Vincent Wesley Couey

Varkonyi and Domokos (2006) proved mono-monostatic convex homogeneous bodies exist, and Sloan (2023) provided analytical equations. However, no verified open geometry exists and the relationship between Sloan's parameterization and actual mono-monostatic behavior has never been computationally tested. We introduce an ECS oracle measuring stable equilibria via drainage basin analysis on the COM height landscape. Applying it to Sloan's parameterization, we find no tested parameter value produces a mono-monostatic body. The surface function has two critical points as proven, but the COM height landscape exhibits 4-11 local minima. Surface critical points are necessary but not sufficient. We resolve this by extending Sloan's phase function with Fourier terms and adding radial perturbations, optimizing via differential evolution. Using both approaches, we construct thirteen verified mono-monostatic bodies (ECS=1 confirmed across merge thresholds 0.5-10%), the first openly published catalog of verified mono-monostatic geometries. The catalog reveals a near-perfect trade-off between self-righting robustness and gentleness (r=0.9993), spanning 7.2x asymmetry range from 0.4% to 3.0% deviation from spherical. Both perturbation families produce overlapping metric profiles, indicating construction method is a convenience not a constraint. All thirteen meshes and construction parameters are openly published.